Question

A woman of mass \(51 \mathrm{kg}\) is standing in an elevator. (a) If the elevator floor pushes up on her feet with a force of $408 \mathrm{N},$ what is the acceleration of the elevator? (b) If the elevator is moving at \(1.5 \mathrm{m} / \mathrm{s}\) as it passes the fourth floor on its way down, what is its speed 4.0 s later?

Short answer

The acceleration of the elevator is approximately -1.80 m/s². The negative sign indicates that the elevator is accelerating downward. (b) What is the elevator's speed 4.0 s after passing the fourth floor? The elevator's speed 4.0 s later is approximately -5.70 m/s. The negative sign indicates that the elevator is moving downward.

Step-by-step solution

Calculate the woman's weight

To calculate the woman's weight, we will use the equation W = mg, where W is the weight, m is the mass (51 kg), and g is the acceleration due to gravity (9.8 m/s²). \(W = 51 \times 9.8\) \(W = 499.8 \mathrm{N}\) The woman's weight is 499.8 N.

Calculate the net force acting on the woman

The net force acting on the woman is the difference between the force exerted by the elevator floor on her feet (408 N) and her weight (499.8 N). \(F_{net} = 408 - 499.8\) \(F_{net} = -91.8 \mathrm{N}\) The net force acting on the woman is -91.8 N. Negative sign indicates that the force is acting downward.

Calculate the acceleration of the elevator

Using Newton's second law of motion (F = ma), we can find the acceleration (a) of the elevator. \(a = \frac{F_{net}}{m}\) \(a = \frac{-91.8}{51}\) \(a \approx -1.80 \mathrm{m/s^2}\) The acceleration of the elevator is approximately -1.80 m/s². The negative sign indicates that the elevator is accelerating downward.

Calculate the elevator's speed 4.0 s later

Now, we need to find the elevator's speed 4.0 s after passing the fourth floor. We will use the equation of motion: v = u + at. We are given the initial velocity (u) as 1.5 m/s (moving downward), the acceleration (a) as -1.80 m/s², and the time (t) as 4.0 s. \(v = 1.5 + (-1.80) \times 4.0\) \(v = 1.5 - 7.20\) \(v = -5.70 \mathrm{m/s}\) The elevator's speed 4.0 s later is approximately -5.70 m/s. The negative sign indicates that the elevator is moving downward.